
Octree subdivision: The general idea of octree (or quadtree in two dimensions) integration is to capture the geometry of a cut element by recursively bisecting sub-cells that intersect with the boundary of the domain, as illustrated in Figure REF . At every level of this recursion, sub-cells that are completely inside the domain are preserved, while sub-cells that are completely outside of the domain are discarded. This cut element subdivision strategy was initially proposed in the context of the finite cell method (FCM) in [1]} and is generally appraised for its simplicity and robustness with respect to cut element configurations. Octree integration has been widely adopted in immersed FEM, see, e.g., [2]}, [3]}, [4]}, [5]}. Various generalizations and improvements to the original octree procedure have been proposed, of which the consideration of tetrahedral cells [6]}, [7]}, the reconstruction of the unfitted boundary by tessellation at the lowest level of bisectioning [5]}, and the consideration of variable integration schemes for the sub-cells [9]}, are particularly noteworthy. Despite the various improvements to the original octree strategy, a downside of the technique remains the number of integration sub-cells (and consequently the number of integration points) that result from the procedure, especially in three dimensions and with high-order bases, where the refinement depth needs to be increased under mesh refinement to reduce the integration error with the same rate as the approximation error [10]}.

Cut element reparameterization: Accurate cut element integration schemes can be obtained by modifying the geometry parameterization of cut elements in such a way that the immersed boundary is fitted. This strategy was originally developed in the context of XFEM by decomposing cut elements into various sub-cells with only one curved side and then to alter the geometry mapping related to the curved sub-cell to obtain a higher-order accurate integration scheme [11]}. This concept has been considered in the context of implicitly defined geometries (level sets) [12]}, [13]}, [14]}, [15]}, the NURBS-enhanced finite element method (NEFEM) [16]}, [17]}, the Cartesian grid finite element method (cgFEM) [18]}, [19]}, and the mesh-transformation methodology presented in [20]}. In the context of the finite cell method, the idea of cut element reparameterization has been adopted as part of the smart octree integration strategy [21]}, [22]}, [23]}, where a boundary fitting procedure is employed at the lowest level of octree bisectioning in order to obtain higher-order integration schemes for cut elements with curved boundaries. Reparameterization procedures have the potential to yield accurate integration schemes at a significantly lower computational cost than octree procedures, but generally compromise in terms of robustness with respect to cut element configurations.

Polynomial integration: Provided that one can accurately evaluate integrals over cut elements (for example using octree integration), it is possible to construct computationally efficient integration rules for specific classes of integrands. In the context of immersed finite element methods, it is of particular interest to derive efficient cut-element integration rules for polynomial functions. The two most prominent methods to integrate polynomial functions over cut elements are moment fitting techniques [24]}, [25]}, [23]}, [27]}, [28]}, in which integration point weights and (possibly) positions are determined in order to yield exact quadrature rules, and equivalent polynomial methods [29]}, [30]}, in which a non-polynomial (e.g., discontinuous) integrand is represented by an equivalent polynomial which can then be treated using standard integration procedures. Such methods have been demonstrated to yield efficient quadrature rules for a range of scenarios. A downside of such techniques is the need for the evaluation of the exact integrals (using an adequate cut-element integration procedure) in order to determine the optimized integration rules. This can make the construction of such quadrature rules computationally expensive, which makes them more suitable in the context of time-dependent and non-linear problems (with fixed boundaries and interfaces), for which the construction of the integration rule is only considered as a pre-processing operation (for each cut element) and the optimized integration rule can then be used throughout the simulation. In the group of Dominik Schillinger at TU Darmstadt, work is currently being done to use neural networks for the computation of integration point weights to accelerate this process.

Dimension-reduction of integrals:
Depending on the problem under consideration, it can be possible to reformulate volumetric integrals over cut elements by equivalent lower-dimensional integrals. This approach is advantageous from a computational effort point of view, as the equivalent integrals are generally less costly to evaluate. A reformulation of volume integrals in terms of boundary integrals has been proposed in the context of XFEM in [31]} and in the immersed FEM setting in [32]}. Another dimension-reduction approach pertains to high-order quadrature for embedded methods with hexahedral background meshes and implicitly defined convex surfaces [33]}. This technique relies on a reduction of integrals to one-dimensional integrations and provides strictly positive weights. The methodology proposed in [34]} (and references therein) provides closed-form formulas for the integration of monomials on convex and nonconvex polyhedra and the extension to curved domains, relying on a reduction of integrals up to vertex evaluations. A downside of dimension-reduction techniques is that they are less general than standard quadrature rules. For example, some techniques can lead to negative quadrature weights in the case of high-order FEM [35]}, which can potentially cause instabilities with inexact arithmetics.

Parameter optimization: Various strategies have been proposed to optimize the parameters of the cut element integration techniques discussed above, most notably for octree subdivision techniques. In [9]}, [10]} algorithms are proposed to select the integration order on the different levels of sub-cells. Ref. [35]} presents a methodology to reduce the number of integration points in a manner similar to moment fitting techniques. These optimization techniques have demonstrated that reducing the number of integration points does not necessarily compromise the accuracy of the simulations. This is theoretically supported by Strang's first lemma [39]}, [40]}, [41]}, which indicates that integration does not need to be exact in order to attain (optimal) convergence [39]}As formulated by Strang and Fix in 1973 [39]}: “What degree of accuracy in the integration formula is required for convergence? It is not required that every polynomial which appears be integrated exactly.". It should be mentioned that this lemma is also considered in the context of CutFEM in e.g., [44]}, [45]}, [10]}.

